homework and exercises - Why is the Dirac mass term Hermitian if Grassmann-valued fields anticommute?

Let's have Dirac mass term in lagrangian:

$$L_{M} = \bar{\Psi}\Psi$$ Lagrangian must be real-valued, i.e., its Hermitian conjugation doesn't change it. But due to Grassmann nature of spinor fields, $[\psi_{a}^{*}, \psi_{b}]_{+} = 0$, $$L_{M}^{\dagger} = -\bar{\Psi}\Psi$$ Where have I make a mistake?